Suppression of resistive interchange instability by external RMP

We experimentally investigate the effect of external resonant magnetic perturbation (RMP) on resistive interchange MHD instability which is typically observed in the Large Helical Device. We apply the m/n = 1/1 external RMP to discharges with the m/n = 1/1 interchange instability. We find that the electron density fluctuation as well as the magnetic field one is clearly reduced without a reduction of the pressure gradient by imposing the external RMP. Moreover, the achieved beta value as well as the beta gradient at the resonant surface is a little improved under certain conditions. Next, we investigate the responses of interchange instability to the external RMP under some different operational conditions. We find that the amplitude of the external RMP to completely suppress the instability with the shielding of the external RMP has a higher correlation with the volume-averaged beta value than other plasma parameters.


Introduction
For the development of economical nuclear fusion reactors with magnetically confined torus plasmas, we should be able to stably confine more than 5% of volume-averaged beta plasmas. However, there are some obstacles to achieve this. MHD instabilities are one of them because they disturb the nested magnetic surfaces, due to the induced plasma current and/or the plasmas escaping from the magnetic bottles, due to the electric field induced in the plasmas. The MHD instability should be avoided by limiting the operational range and/or suppressed by certain methods, because it degrades the plasma * Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. confinement performance. As the most probable candidates for a magnetically confined torus fusion reactor, there are two types of tori, tokamaks and helical systems, which are different in their methods to generate magnetic surfaces, which lead to differences in the mainly induced MHD instability.
In tokamaks, current driven MHD instabilities such as the tearing mode (including the locked mode and the neoclassical tearing mode), the resistive wall mode (RWM), and the edge localized mode (ELM) are dominant, because large plasma current exists to generate the magnetic surfaces. The instabilities not only degrade the plasma confinement performance through a decrease of the beta-value, but also lead to rapid discharge termination and cause the device itself excessive damage. In tokamaks, much research on the active control of instabilities has been done because the impact of MHD instabilities on plasma confinement is serious, as above-mentioned. Imposing resonant magnetic perturbation (RMP) by external coils, the control of plasma flow, and pellet injection have been proposed as control methods, and their effectiveness is already known. Examples of suppressing instabilities by applying the external RMP are mentioned as follows. For the tearing mode and/or the locked mode, applying rotating RMP suppresses a collapse by keeping the mode rotation and preventing a locking of the mode [1]. For the RWM, the applied external RMP directly suppresses magnetic fluctuation around a plasma boundary [2]. For the ELM, applying the external RMP suppresses onsets of the ELM by disturbing the magnetic surfaces in the pedestal region and control of the pressure gradient [3].
On the other hand, the helical system does not basically need a large plasma current, unlike tokamaks, and some like the Large Helical Device (LHD) [4] have a magnetic hill region. Then, pressure driven instabilities are dominant. Especially, the resistive interchange instability is major disequilibrium in the LHD [5]. According to an LHD experimental study [6], resistive interchange instabilities degrade the beta value by almost 5%, and do not induce rapid discharge termination like tokamaks. In helical systems, there is a little research on the active control of MHD instabilities, where it is shown that the amplitude of the resonant magnetic fluctuation decreases with the increment of the external RMP, because the impact of the instabilities on the plasma confinement is not serious, as above-mentioned [7]. However, when we design a helical fusion reactor with more than 5% volume-averaged beta plasmas, we expect a possibility that the interchange instability in the peripheral region prevents greater improvement of the plasma confinement performance of an LHD type reactor. Thus, we should develop an active control method which suppresses the interchange instability and/or prevents a decrease of the beta value.
The purpose of this study is to develop an active control method which suppresses resistive interchange instability and/or prevents a decrease of the beta value in helical plasmas. We apply to the LHD plasmas the control method by external RMP, which has shown its effectiveness in suppressing current driven MHD instabilities in tokamaks, and investigate the effects on the resistive interchange instability. Because the m/n = 1/1 resonant surface exists at the peripheral region in a typical magnetic configuration of the LHD, and it is in the magnetic hill region, the target to suppress in this study is the interchange instability with m/n = 1/1. Here, 'm' and 'n' indicate poloidal and toroidal mode numbers, respectively. Specifically, we apply the m/n = 1/1 external RMP, whose amplitude changes with every discharge, to those with the m/n = 1/1 resistive interchange instability in the LHD, and we investigate the response of interchange instability to the external RMP, under various operational conditions. This paper is organized as follows. In section 2, we describe the experimental set-up. In section 3, a typical response of the resistive interchange instability to the external RMP is shown. In section 4, we show how the response shown in section 3 changes when the operational magnetic field and/or the volume-averaged beta value are changed. In section 5, we show whether or not the response shown in section 3 changes when the way to apply external RMP is changed. Finally, we give a brief summary.

Experimental set-up
The LHD is a heliotron device, which is a type of helical system. It has a pair of helical coils with three layers, and three pairs of poloidal coils. As a typical device specification, the major and the minor radius (R ax and a p ) are 3.5-4.0 m and 0.46-0.64 m, respectively. In the LHD, we can control the rotational transform profile (magnetic shear) and the height of the magnetic hill through changing the torus major radius location of the magnetic axis, the shape of plasma poloidal crosssection and the plasma aspect ratio, which are changed through the control of the current ratio between poloidal coils and/or between three layers of helical coils [8]. In this study, we investigate how the m/n = 1/1 resistive interchange instability responds to the external RMP for the discharges in which the R ax is 3.75 m, the toroidally averaged plasma cross-section shape is almost circular, and the plasma aspect ratio is 6.3. In the above experimental condition, the magnetic surface of ℩ = 1 is located at r/a p ∼ 0.9 in the vacuum and it always exists in the magnetic hill region during discharges. Here, 'r' is a minor radius of the plasma. Figure 1 shows typical waveforms of a discharge with resistive interchange instability without imposing the external RMP. The operational magnetic field strength at the magnetic axis is 1.2 T. Figure 1(a) shows the time evolution of the volume-averaged beta value, determined by diamagnetic measurement, the total port-through power of tangentially injected neutral beam (NB) and electron cyclotron heating (ECH). The plasma is initiated by ECH, and maintained by balanced neutral beam injection (NBI) from t = 3.3 s to 6.3 s. Figure 1(b) shows the time evolution of the line-averaged electron density measured by far-infrared (FIR) laser interferometer and the total port-through injection power of the two beam lines of the perpendicular NB, which is injected as a probe beam to measure the plasma flow speed. Figure 1(c) shows the time evolution of the plasma current normalized by the operational magnetic field, and the plasma current at the termination of the tangential NBIs corresponds to the increment of the rotational transform at the edge by almost 0.04. Most of the plasma current is expected to be induced by bootstrap current, not by Okawa-current, because the tangential NBIs are almost balanced. Figure 1(d) shows the time evolution of the poloidal flow speed around the ℩ = 1 surface. The poloidal flow is measured by charge exchange spectroscopy [9]. Figures 1(e) and (f ) show the time evolution of the m/n = 1/1 amplitude of the poloidal field fluctuation measured by the magnetic probe, and the m/n = 1/1 electron density fluctuation amplitude around the ℩ = 1 surface measured by CO2 laser interferometer, respectively. Fluctuation amplitude in figures 1(e) and (f ) corresponds to the RMS during 20 ms. We measure the magnetic field fluctuation by multiple magnetic probes installed outside the plasma. On the other hand, the amplitude of the electron density fluctuation measures the response of the local fluctuation around the resonance surface. It should be noticed that the yellow shadow region in figure 1 shows the time window for the analysis, when the plasma is in an almost steady state. According to previous research, the fluctuation shown in figures 1(e) and (f ) is expected, due to resistive interchange instability [5,6]. The RMP coil system in the LHD consists of ten pairs of square shape coils installed at the top and bottom of the torus, and it generates the resonant magnetic field mainly with the m/n = 1/1 and/or 2/1 in the plasma region by using three power supplies [10]. In the RMP system of the LHD, the external RMP does not rotate, and its amplitude is kept constant in time, or increases or decreases at a constant rate of change during discharges. In sections 3 and 4 of this paper, its amplitude is kept constant. On the other hand, in section 5, its amplitude increases during discharges. In sections 3-5, we use the maximum current value of the external RMP coils as an amplitude of the imposed external RMP [11]. The 1.11 kA T −1 as the coil current is expected through calculation to induce a magnetic island with a width of 20% of the plasma minor radius in the vacuum. The width is consistent with that of the vacuum magnetic surface measurement [10].
When we investigate the response of the instability to the external RMP, we focus on the following four parameters; (1) the m/n = 1/1 magnetic field fluctuation amplitude, (2) the electron density fluctuation amplitude which has the high coherence with the m/n = 1/1 magnetic field fluctuation at the ℩ = 1 surfaces, (3) the pressure gradient at the ℩ = 1 surface, and (4) the poloidal flow speed at the ℩ = 1 surface. In this study, we consider the magnetic field and the electron density fluctuation amplitude as an index of the unstable level of the interchange instability.
The unstable level of interchange instability is strongly affected by a level of the pressure gradient, magnetic shear, and the magnetic hill height around the resonant surface. In the experiment sequences to investigate the instability response to the external RMP, the heating and fueling conditions, and the magnetic configuration are exactly same without the amplitude of the external RMP imposed. Actually, from the time evolution of the plasma current, the line-averaged electron density and the volume-averaged beta value during the experiment sequence (see figure 2), they change little, which leads to a small difference in magnetic shear and the magnetic hill height before the external RMP penetrates into the resonant surface, as shown by the left area of the range painted blue in figure 2. It should be noted that the external RMP penetration includes the meaning of both 'full-penetration' and 'partial-penetration', and a flattening of the electron temperature is observed in each case. So, we recognize a flattening of the electron temperature as a state of the external RMP penetration. Moreover, a flattening of the electron temperature is evaluated from the pressure gradient, because a sudden decrease of the pressure gradient and beta is observed due to the external RMP penetration [12,13]. Then, we mainly focus on the change of pressure gradient in this paper. The pressure profile is evaluated from the electron temperature profiles by Thomson scattering measurement and the line integrated electron density profile measured by FIR. Here, the radial resolution of the temperature profile measurement is less than ∆ρ(normalized minor radius) = 0.02, and that of the density profile is less than ∆ρ = 0.1. The temperature profile is much more sensitive to the existence of the magnetic island than the density profile, and the radial resolution of the temperature profile is much less than the width of the typical magnetic island when the external RMP penetration [12]. We need equilibrium data to know the relationship between a normalized plasma minor radius (ρ ≡ r/a p ) and the real coordinates. We select an equilibrium where the torus inboard electron temperature best fits the outboard one in an equilibrium database, which is calculated by the three-dimensional MHD equilibrium analysis code (VMEC), with various pressures and their profiles under no plasma current. The local electron density radial profile is evaluated by Abel inversion. Here we assume n i = n e , T i = T e , and Z eff = 1. When we evaluate the pressure gradient around the ℩ = 1 surface, we use the electron temperature profile averaged between the torus-inside and outside data. It should be noted that the ℩ = 1 surface is identified from the flattening location of the electron temperature when the external RMP with m/n = 1/1 penetrates into the resonant surface, and its location is assumed to be same with that in the case of the shielding of the RMP.
According to some theoretical research, the poloidal flow speed around the resonant rational surface affects the stability of interchange instability [14,15]. We observe the poloidal flow speed response to the external RMP. Figure 3 shows how the interchange instability responds to the external RMP under a discharge condition where the operational field strength is 1.2 T and the volume-averaged beta value is 0.85%. The detailed operational conditions are shown in table 1. In these experiment sequences, we conducted ten discharges, and the external RMP amplitude was changed with  zero. The reason is that it is more difficult to extract the coherent electron density fluctuation than the coherent magnetic fluctuation. Figure 3(c) shows the absolute value of the beta gradient at the ℩ = 1 surface as a function of the external RMP coil current. Each symbol in figure 3(c) indicates the timeaveraged absolute value of the beta gradient during the time window for the analysis, and its error bar indicates the square root of the variance. The beta gradient in the region, when the external RMP coil current is 0.93 kA T −1 or less, is kept almost equal to or greater than that in the case of no external RMP, indicated by the purple dashed line in the graph. Though the interchange instability has the property that it becomes more unstable as the gradient increases, because it is a pressure driven instability, the fluctuation amplitude decreases without decrease of the pressure gradient. Moreover, the pressure gradient tends to increase as the external RMP coil current increases, which suggests the suppression of instability and the small improvement of the plasma confinement performance by imposing the external RMP. The increasing tendency of the volume-averaged beta value with the external RMP less than 0.93 kA T −1 as shown in figure 2(a) also supports the small improvement of the plasma confinement performance. Over 1.1 kA T −1 of the external RMP coil current, the pressure gradient discontinuously decreases due to the external RMP penetration, and the coherent fluctuations disappear. Here, the reason, that the pressure gradient does not decrease to zero even if the magnetic island appears, is that the electron temperature profile is averaged between the torus-inside and outside data in evaluation the beta gradient. We guess that the external RMP penetration into the m/n = 1/1 resonant surface induces the decrease of the pressure gradient through the formation of a magnetic island beyond the RMP penetration, and leads to stabilize the interchange instability. From the above result, it is found that the application of the external RMP with an appropriate strength can suppress the interchange instability, keeping the pressure gradient at the resonant surface. Figure 3(d) shows the poloidal flow speed at the ℩ = 1 surface as a function of external RMP coil current. Each symbol in figure 3(d) indicates the time-averaged value of the poloidal flow speed during the time window for the analysis, and its error bar indicates the time-averaged value of the measurement error. When the external RMP is less than 0.93 kA T −1 , the poloidal flow speed is kept at almost the same value. When it reaches 1.1 kA T −1 , the absolute value of the poloidal flow speed decreases discontinuously to almost zero, which is due to the RMP penetration into the resonant surface and the formation of the magnetic island, as well as the sudden reduction of the plasma pressure gradient, as shown in figure 3(c). The above results suggest that the change of the pressure gradient and the poloidal flow speed does not directly affect the stabilization of the interchange instability. We guess that the change of magnetic field at the plasma boundary due to the applied external RMP would lead to stabilization of the interchange instability because the m/n = 1/1 interchange instability is suppressed with the shielding of the external RMP in the experiments. It is one of our future subjects to make it clear that the change of the magnetic field at the boundary, due to the application of the external RMP, leads to the stabilization of the interchange instability.

Dependence of the interchange instability response to the external RMP on the discharge conditions
As mentioned in section 3, it is found that the fluctuation amplitude due to the instability decreases as the external RMP increases under an operational condition. In this section, we investigate the response of the resistive interchange instability to the external RMP under various discharge conditions. We use the magnetic fluctuation amplitude as a stabilization index because it is more difficult to extract the coherent density fluctuation than the magnetic fluctuation, as mentioned in section 3. Figures 4 and 5 show the responses of (a) the m/n = 1/1 magnetic field fluctuation amplitude and (b) the beta gradient at the resonant surface to the external RMP for different discharge conditions. Figure 4 corresponds to a discharge where the operational magnetic field strength is 1.2 T and the volumeaveraged beta value is 0.85%, which is exactly same with figures 3(a) and (c). Figure 5 corresponds to a discharge where the field strength is 0.9 T and the volume-averaged beta value is 1.3%. In figure 4, the instability is completely suppressed when the external RMP coil current is about 1.0 kA T −1 , indicated by the red dashed line in the graph. This current value is almost the same as the penetration threshold of the external RMP, estimated from the discontinuous decrease of the pressure gradient (0.93-1.1 kA T −1 ). On the other hand, in figure 5, when the external RMP is about 1.6 kA T −1 , the instability is suppressed. However, there the pressure gradient and the fluctuation amplitude decrease discontinuously, which suggests that the suppression of the instability is due to the external RMP penetration into the resonant surface. If the RMP penetration threshold was larger than 1.6 kA T −1 , the instability would be completely suppressed when external RMP coil current is about 2.0 kA T −1 , indicated by the red dashed line in the graph. The 2.0 kA T −1 is estimated from the extrapolated value of the dependence of magnetic fluctuation amplitude on external RMP coil current. The above result shows that the amplitude of the external RMP to completely suppress the instability has a different dependence from that of the penetration threshold of the RMP. In order to suppress the resistive interchange instability and improve the confinement performance, we need to achieve a complete suppression of the instability with the shielding of the external RMP. As the first step, in this study, we focus on the amplitude of the external RMP to completely suppress the instability. On the parameter dependence study of the RMP penetration threshold in the LHD, there are other works [13,16]. On the parameter dependence study of the complete suppression of the instability, with the shielding of the external RMP, is one of the future subjects. Here it should be noticed that in 0.9 T discharges, the clear improvement of the pressure gradient with the reduction of the magnetic fluctuation amplitude is not observed. The data indicated by the triangles in figure 5 have a little lower electron density than the other data with the shielding of the external RMP, which is a candidate to explain the reason why the clear improvement of the confinement is not observed in the 0.9 T discharges. Figure 6 shows the dependences of the amplitude of the external RMP to completely suppress the instability on (a) the operational magnetic field, (b) the volume-averaged beta value, (c) the line-averaged electron density, and (d) the magnetic field fluctuation amplitude when the external RMP is not applied such as the value of b∼/B t at I RMP /B t = 0 in figure 4(a) or 5(a). In figure 6, the red symbols indicate the case where the amplitude of the external RMP to completely suppress the instability is less than that of the RMP penetration, and the blue ones indicate the case where the amplitude of the external RMP to completely suppress the instability is greater than that of the RMP penetration. It should be noted that the amplitude of the external RMP to completely suppress the instability is estimated from the extrapolated value of the dependence of the magnetic fluctuation amplitude on external RMP coil current. From the above analysis, the amplitude of the external RMP to completely suppress the instability has the highest correlation with the beta value shown in figure 6(b). However, data in figure 6(b) has various discharge conditions in the operational magnetic field and/or collisionality. In order to construct an empirical scaling law of the amplitude of the external RMP to completely suppress the instability, we need a dependence on the beta with the same magnetic field strength/collisionality, and dependence on the collisionality with the same beta and so on, which is one of our future tasks.

Dependence of interchange instability response on patterns of applying external RMP
In this section, we show how the patterns of applying the external RMP affect the interchange instability response. We compare the instability responses between the constant RMP in time and the ramping up of RMP during the discharges. Dependences of the amplitude of the external RMP to completely suppress the instability with the shielding of the external RMP on (a) the operational magnetic field, (b) the volume-averaged beta value, (c) the line-averaged electron density, and (d) the magnetic field fluctuation amplitude when the external RMP is not applied. The red symbols indicate the case where the amplitude of the external RMP to completely suppress the instability is less than that of the RMP penetration, and the blue ones indicate the case where the amplitude is greater than that of the RMP penetration.   Figure 8 shows the responses of (a) the m/n = 1/1 magnetic field fluctuation amplitude and (b) the beta gradient at the resonant surface to the external RMP, in the case of keeping the RMP coil current constant in time (black symbols, as shown in figure 7(a)) and the case of ramping it up (red lines, as shown in figure 7(b)) during discharges. In figure 8, each red line corresponds to one discharge and is made by connecting with straight lines, multiple 0.1 s interval data during a constant NBI heating condition. Figure 8 shows that the responses of the magnetic field fluctuation amplitude and the pressure gradient to the ramping up of external RMP are quantitatively the same as that of constant external RMP, which suggests that the patterns of applying the external RMP do not affect the interchange instability response.

Conclusion
We experimentally investigate the effect of the external RMP on the resistive interchange MHD instability which is typically observed in the LHD. We apply the m/n = 1/1 external RMP to the discharges with the m/n = 1/1 interchange instability. It should be noted that the resonant surface of the instability is located at the peripheral region, and the instability is an expected obstacle to achieve a high beta operation equivalent to nuclear fusion reactors in the future. In helical systems, a little research on the active control of MHD instabilities has been done. In this paper, we newly find the following properties.
The electron density fluctuation as well as the magnetic field one is clearly suppressed without a reduction of the pressure gradient by imposing the external RMP. Moreover, a little improvement of the achieved beta value, as well as the beta gradient at the resonant surface, is observed, depending on the experiment condition. In order to investigate the stabilizing mechanism of the interchange mode by the external RMP, we investigate the poloidal flow speed at the resonant surface. The change of the poloidal flow speed due to applying the RMP is not the cause, because it hardly changes before the instability suppression due to the RMP. To resolve the stabilization mechanism is one of our future subjects.
Next, we investigate the responses of the interchange instability to the external RMP under some different operational conditions, like the operational magnetic field strength, the volume-averaged beta value and/or the line-averaged electron density and so on. In some discharge conditions, the external RMP penetrates into the resonant surface before the complete suppression of the instability. Here, we focus on the amplitude of the external RMP to completely suppress the instability, which is estimated from the extrapolated value of dependence of magnetic fluctuation amplitude on the external RMP coil current. We find that the amplitude has a higher correlation with the volume-averaged beta value than with other plasma parameters.
Finally, in order to investigate the effect of the way to apply the external RMP, we compare two patterns of this; one is when the RMP is kept constant in time, and the other is when the RMP ramps up during discharges. We find that the response of the resistive interchange instability and the pressure gradient to the ramping up of external RMP is quantitatively the same as that of constant external RMP, which suggests that the patterns of applying external RMP do not affect the interchange instability response.